# Working Both Sides

## Abstract

Accepting the importance of Jim Greeno’s work, not just in this volume but throughout his career, I offer this commentary from the position of a researcher who first worked “from the inside out” and who now works “from the outside in.” My *identity* is that of a mathematics educator with a theoretical commitment to design research. To clarify, for almost seven years I collaborated with Paul Cobb, Koeno Gravemeijer and others in the execution of classroom design experiments in which I acted as the teacher. In these settings, I was working from the inside out to first, *in action*, make sense of students’ understandings so that I could planfully orchestrate classroom discussions. Later, I would conduct retrospective analyses of my interactions by analyzing from the “outside” what I had previously participated in on the “inside.” In these instances, I worked to understand both the students’ and my learning through normative patterns of engagement. The theoretical lens that I adopted for most of my analyses of the classroom is that of a social constructivist with a strong emphasis on tools. I find Greeno’s levels of accounts of cognition in interaction strengthen my previous orientation by more clearly articulating levels of a progression of conceptual understanding. However, I am left wondering what the means of support are for shifts between the levels. Clearly, having a way to analyze the students’ current abilities or ways of reasoning is crucial. However, I view it as necessary but insufficient for supporting learning. It is this stance that I take in my commentary.

### Keywords

Coherence Liner Stein Clarification## Notes

### Acknowledgements

I would like to thank the teachers in the Madison School District who participate in the Vanderbilt Teacher Collaborative at Madison [http://www.vtcm.org]. The analysis reported in this paper was supported by the National Science Foundation under grant REC-0135062.

### References

- Ball, D. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics.
*The Elementary School Journal, 93*, 373–397.CrossRefGoogle Scholar - Ball, D., & Cohen, D. (1999). Developing practice, developing practitioners: Towards a practice-based theory of professional education. In G. Sykes & L. Darling-Hammond (Eds.),
*Teaching as the learning profession: Handbook of policy and practice*(pp. 3–32). San Francisco: Jossey-Bass.Google Scholar - Brown, A. L. (1992). Design experiments: Theoretical and methodological challenges in creating complex interventions in classroom settings.
*Journal of the Learning Sciences, 2*, 141–178.CrossRefGoogle Scholar - Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research.
*Educational Researcher, 32*(1), 9–13.CrossRefGoogle Scholar - Cobb, P., & McClain, K. (2001). An approach for supporting teachers’ learning in social context. In F. -L. Lin & T. Cooney (Eds.),
*Making sense of mathematics teacher education*(pp. 207–232). Dordrecht: Kluwer Academic.Google Scholar - Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research.
*Educational Studies in Mathematics, 30*, 458–477.CrossRefGoogle Scholar - diSessa, A., & Cobb, P. (2004). Ontological innovation and the role of theory in design experiments.
*The Journal of the Learning Sciences, 13*, 77–104.CrossRefGoogle Scholar - Franke, M. L., Carpenter, T. P., Levi, L., & Fennema, E. (1998, April).
*Capturing teachers’ generative change: A follow-up study of teachers’ professional development in mathematics.*Paper presented at the annual meeting of the American Educational Research Association, San Diego.Google Scholar - Franke, M. L., & Kazemi, E. (2001). Teaching as learning within a community of practice: Characterizing generative growth. In T. Wood, B. Nelson, & J. Warfield (Eds.),
*Beyond classical pedagogy in elementary mathematics: The nature of facilitative teaching*(pp. 47–74). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Kaput, J. J. (1994). The representational roles of technology in connecting mathematics with authentic experience. In R. Biehler, R. W. Scholz, R. Strasser, & B. Winkelmann (Eds.),
*Didactics of mathematics as a scientific discipline*. Dordrecht: Kluwer.Google Scholar - Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching.
*American Educational Research Journal, 27*, 29–63.Google Scholar - Lampert, M. (2001).
*Teaching problems and the problems in teaching*. New Haven, CT: Yale University Press.Google Scholar - Latour, B. (1987).
*Science in action: How to follow scientists and engineers through society*. Cambridge, MA: Harvard University Press.Google Scholar - McClain, K. (2002a). A methodology of classroom teaching experiments. In S. Goodchild & L. English (Eds.),
*Researching mathematics classrooms: A critical examination of methodology*(pp. 91–118). Westport, CT: London, Praeger.Google Scholar - McClain, K. (2002b). Teacher’s and students’ understanding: The role of tools and inscriptions in supporting effective communication.
*Journal of the Learning Sciences, 11*, 217–249.CrossRefGoogle Scholar - McClain, K., & Cobb, P. (1998). The role of imagery and discourse in supporting students’ mathematical development. In M. Lampert & M. Blunk (Eds.),
*Mathematical talk and school learning: What, why, and how*(pp. 56–81). Cambridge: Cambridge University Press.Google Scholar - McClain, K., & Schmitt, P. (2004). Extending teachers’ mathematical understandings: A case from statistical data analysis.
*Mathematics Teaching in the Middle Schools, 9*, 274–279.Google Scholar - McClain, K., Zhao, Q., Visnovska, J., & Bowen, E. (2009). Understanding the role of the institutional context in the relationship between teachers and text. In J. T. Remillard, B. Herbel-Eisenmann & G. Lloyd (Eds.),
*Mathematics teachers at work: Connecting curriculum materials and classroom instruction*. (pp. 56–69). New York: Routledge.Google Scholar - Meira, L. (1995). The microevolution of mathematical representations in children’s activitiy.
*Cognition and Instruction, 13*, 269–313.CrossRefGoogle Scholar - Meira, L. (1998). Making sense of instructional devices: The emergence of transparency in mathematical activity.
*Journal for Research in Mathematics Education, 29*, 121–142.CrossRefGoogle Scholar - Pickering, A. (1995).
*The Mangle of Practice.*Chicago: The University of Chicago Press.Google Scholar - Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective.
*Journal for Research in Mathematics Education, 26*, 114–145.CrossRefGoogle Scholar - Simon, M. (1997). Developing new models of mathematics teaching. In E. Fennema & B. S. Nelson (Eds.),
*Mathematics teachers in transition*(pp. 55–86). Mahwah, NJ: ErlbaumGoogle Scholar - Steffe, L. P., & Cobb, P. (1988).
*Construction of arithmetical meanings and strategies*. New York: Springer.Google Scholar - Thompson, P. W. (2002). Didactic objects and didactic models in radical constructivism. In K. Gravemeijer, R. Lehrer, B. v. Oers, & L. Verschaffel (Eds.),
*Symbolizing, modeling and tool use in mathematics education*(pp. 197–220). Dordrecht: Kluwer.Google Scholar - van Oers, B. (1996). Learning mathematics as meaningful activity. In P. Nesher, L. Steffe, P. Cobb, G. Goldin, & B. Greer (Eds.),
*Theories of mathematical learning*(pp. 91–114). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar - van Oers, B. (2000). The appropriation of mathematical symbols: A psychosemiotic approach to mathematical learning. In P. Cobb, E. Yackel, & K. McClain (Eds.),
*Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design*(pp. 133–176). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Varela, F. J., Thompson, E., & Rosch, E. (1991)
*The embodied mind: Cognitive science and human experience.*Cambridge, MA: MIT Press.Google Scholar